Research Statement of Michael J. Kozdron

Research Statement of Michael J. Kozdron

Mathematics Subject Classification: 60, 82.

My research is in probability, and is an offshoot from work that arose in the study of the Schramm-Loewner evolution (SLE_k).

A very general goal in statistical physics is to understand the behaviour of a system at criticality; that is, at or near the temperature at which there is a phase transition. In a number of instances the physical system is well-described by a discrete, or lattice, model, and many two-dimensional lattice models are conjectured to have a conformally invariant scaling limit. By assuming conformal invariance, it is often possible to give exact predictions for certain quantities known as critical exponents which describe the qualitative behaviour of the systems. The most basic example is that of simple random walk which, appropriately normalized, converges to Brownian motion. Other examples include loop-erased random walk, domino tilings, uniform spanning trees, percolation, and the Ising and Potts models. The self-avoiding walk, a model of polymer chains introduced by the chemist P. Flory in the 1940's, is a model where minimal rigorous progress has been made, and is one of the motivations for much of this entire program of study. My particular area of specialization in this program is two-dimensional excursion theory.

Historical Background and Research Motivation

One-dimensional Brownian motion excursions were formally introduced by K. Itô in 1971, though they had been understood much earlier by P. Lévy. The Poisson-point-process view has successfully been exploited by a number of researchers as indicated in Rogers-Williams [8].

In contrast, the theory of multi-dimensional Brownian excursions is much more incomplete, and they were neglected until 1984. In his book [1], K. Burdzy introduces excursion laws in a domain from a distinguished boundary point. His motivation was to study the properties of the initial part of the excursion, and he restricted attention primarily to Lipschitz domains since he could give local properties of excursions in explicit form only for these particular regions.

The point of view of two-dimensional Brownian excursion measures in an arbitrary simply connected domain in C taken in my dissertation work was introduced in Lawler-Werner's 2000 paper [6] on Brownian intersection exponents. They discovered that certain critical exponents could be computed exactly from Brownian excursions in a rectangle.

In 2001, S. Fomin [2] proved an identity relating loop-erased random walk probabilities to determinants of matrices of hitting measures. He showed that the probability that a first random walk starting at x¹ exits a domain at y¹, and that a second random walk starting at x² exits the domain at y² and avoids the loop-erasure of the first path, is given by the determinant of the hitting matrix. Actually, he proved this identity in general for the loop-erasure of discrete stationary Markov chains, and conjectured that it held for continuous processes as well, provided that the model was discretized and appropriate limits taken.

Past Work

In my dissertation [3] I relate the work of Lawler-Werner with that of Fomin. I complete a thorough investigation of Brownian excursion measure in C, developing it in a more general topology than originally done by Lawler-Werner. I also prove that the discrete analogue, simple random walk excursion measure, converges to Brownian excursion measure for domains with rough boundaries. It is important for applications that the boundary of the domain can be at least as rough as the excursion path, and my dissertation result holds for any simply connected domain whose boundary is a Jordan curve. This proves, and extends, the claim in [6] that such was true. It must be noted that because conformal invariance and the Riemann mapping theorem are crucial tools in my work, these results are exclusively two-dimensional.

While Brownian excursion measure $\mu_{\partial D, x,y}(\cdot)$ is the measure on excursions $\omega \in \Omega$ from $x \in \partial D$ to $y \in \partial D$ , its mass $\mu_{\partial D,x,y}(\Omega) = H_{\partial D}(x,y)$ can be thought of as the hitting measure of an excursion. Hence, the continuous analogue of Fomin's determinant is the determinant of the matrix of excursion hitting measures, $\det [H_{\partial D}(x^i,y^j)]_{1\le i,j \le n}$ . I show that both excursion measure (extending the proof in [6] to the more general topology) and the determinant of the excursion hitting matrix are conformally invariant. This then shows that two discrete models, namely random walk excursion measure and Fomin's determinant for random walk, have conformally invariant scaling limits.

Future Work

Although the loop-erasing procedure is well-defined for discrete processes, there is no analogous notion of "loop-erased Brownian motion" since loops exist at arbitrarily small scales. Lawler-Schramm-Werner [5] recently showed that the scaling limit of loop-erased random walk in the plane is SLE₂. In mid-2003, Lawler-Werner [7] introduced the Brownian loop soup in which a Poissonian cloud of bubbles is added to an SLE₂ path to produce a Brownian motion. To date, SLE₂ is the best possibility for what might be termed "loop-erased Brownian motion."

In [4], it is shown that in Fomin's identity the determinants of the random walk hitting matrices converge in the scaling limit to the determinant of the Brownian excursion hitting matrix. In the future, I would like to study the other part of Fomin's identity, and try to show directly that the probability that a Brownian excursion and an SLE₂ each starting and ending at specified points do not intersect is the determinant of the excursion hitting matrix.

Bibliography

[1]: Krzysztof Burdzy.
Multidimensional Brownian excursions and potential theory.
Longman Scientific & Technical, Guildford, UK, 1987.
[2]: Sergey Fomin.
Loop-erased walks and total positivity.
Trans. Amer. Math. Soc., 353:3563-3583, 2001.
[3]: Michael J. Kozdron.
Simple random walk excursion measure in the plane.
Ph.D. dissertation, Duke University, 2004.
[4]: Michael J. Kozdron and Gregory F. Lawler.
Estimates of random walk exit probabilities and application to loop-erased random walk.
Preprint, 2005. Available at arXiv:math.PR\0501189.
[5]: Gregory F. Lawler, Oded Schramm, and Wendelin Werner.
Conformal invariance of planar loop-erased random walks and uniform spanning trees.
Ann. Probab., 32:939-995, 2004.
[6]: Gregory F. Lawler and Wendelin Werner.
Universality for conformally invariant intersection exponents.
J. Eur. Math. Soc., 2:291-328, 2000.
[7]: Gregory F. Lawler and Wendelin Werner.
The Brownian loop soup.
Probab. Theory Related Fields, 131:565-588, 2004.
[8]: L. C. G. Rogers and David Williams.
Diffusions, Markov Processes, and Martingales. Volume 2: Itô Calculus.
Cambridge University Press, Cambridge, UK, second edition, 2002.

Michael's Home Page
February 13, 2005